C2 chapter 6 - radian measure and its applications

In this chapter you will learn at least one new way to measure angles and you'll hopefully appreciate why this new way makes sense.


C2 section 6.1

You need to have a good working knowledge of measuring angles in degrees, know the key names for parts of circles and be able to recall and apply the formulas for circumference and arc length.
We are learning to: redefine how we measure angles and use this to calculate arc lengths for sectors.

Here's the Khanacademy introduction to radians. Have a watch and see if that makes sense.




So the idea is that rather than putting 360° equal to a full turn, you define a full turn to be 2π radians. Now you might well ask "why 2π radians? Wouldn't it make sense to define a full turn some other way?" This is an excellent question, but some people have got there before you:





Vi Hart makes an impassioned plea for us to replace π with τ here:





... And you'll have fun with her 'Song about a Circle Constant' below:





There's more Tau versus Pi debate to be had at Tau Day

Now, let's not get confused here. The examiner in a C2 exam isn't going to give you credit for answers written in terms of τ. But thinking in terms of τ actually turns (pun not intended!) out to be a great idea... especially as we start to learn about radians. So here we go:

C2 Exercise 6A

• C2 Exercise 6A worked solutions:

• Questions similar to this exercise:
• degrees to radians questions in khanacademy
• radians to degrees questions in khanacademy
• and radians and degrees both ways in khanacademy

C2 Section 6.2

LO: WALT calculate arc length and solve worded problems involving arc length

C2 Exercise 6B

• C4 Exercise 6B worked solutions:

C2 exercise 6B (arc length).pdf

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C2 exercise 6B (arc length).jnt

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C2 Section 6.3

LO: WALT calculate sector area of sectors defined by angle θ in radians and solve worded problems involving sector areas.
Since the area of a circle is given by

$\displaystyle A = \pi r^2$

The area of a sector with angle θ is the fraction θ out of 2π of the area of the circle hence for a sector of angle θ and radius r,

$\displaystyle A = \dfrac {\theta} {2\pi} \pi r^2$

which simplifies to

$\displaystyle A = \dfrac {1} {2} r^2 \theta$

C2 Section 6.4

LO: WALT calculate sector area of segments defined by angle θ in radians and solve worded problems involving segment areas.

Since the area of a sector is given by

$\displaystyle A = \dfrac {1} {2} r^2 \theta$

and the area of the triangle defined by the same angle is

$\displaystyle A = \dfrac {1} {2} r^2 \sin{\theta}$

The area of a segment with angle θ is the area of the sector subtract the area of the matching triangle:

$\displaystyle A = \dfrac {1} {2} r^2 (\theta - \sin{\theta})$

C2 Exercise 6C

• C2 Exercise 6C worked solutions (covers sections 6.3 and 6.4):